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Home » Issues » Archive of Issues » 2021 - 4 issue » pp. 519-535

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Uzbekistan Tashkent
Year
2021
Volume
31
Issue
4
Pages
519-535
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Section Mathematics
Title Structure of singular sets of some classes of subharmonic functions
Author(-s) Abdullaev B.I.a, Imomkulov S.A.b, Sharipov R.A.a
Affiliations Urgench State Universitya, Institute of Mathematics, National Academy of Sciences of Uzbekistanb
Abstract In this paper, we survey the recent results on removable singular sets for the classes of $m$-subharmonic ($m-sh$) and strongly $m$-subharmonic ($sh_m$), as well as $\alpha$-subharmonic functions, which are applied to study the singular sets of $sh_{m}$ functions. In particular, for strongly $m$-subharmonic functions from the class $L_{loc}^{p}$, it is proved that a set is a removable singular set if it has zero $C_{q,s}$-capacity. The proof of this statement is based on the fact that the space of basic functions, supported on the set $D\backslash E$, is dense in the space of test functions defined in the set $D$ on the $L_{q}^{s}$-norm. Similar results in the case of classical (sub)harmonic functions were studied in the works by L. Carleson, E. Dolzhenko, M. Blanchet, S. Gardiner, J. Riihentaus, V. Shapiro, A. Sadullaev and Zh. Yarmetov, B. Abdullaev and S. Imomkulov.
Keywords subharmonic functions, $m$-subharmonic functions, strongly $m$-subharmonic functions, $\alpha$-subharmonic functions, Borel measure, $C_{q,s}$-capacity, polar set
UDC 517.559, 517.57
MSC 32U30, 31C05
DOI 10.35634/vm210401
Received 14 July 2021
Language Russian
Citation Abdullaev B.I., Imomkulov S.A., Sharipov R.A. Structure of singular sets of some classes of subharmonic functions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 4, pp. 519-535.
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